Two circles are congruent circles if and only if they have the same radius.
Consider two congruent circles ⊙A and ⊙B, and a point on each one.
Because ⊙A≅⊙B, the distance from the center of the circle to a point on the circle is the same for each circle. Therefore, AC≅BD which implies that both circles have the same radius.
Conversely, consider two circles with the same radius.
Because a rigid motion maps one circle onto the other, it is concluded that both circles are congruent.